Orthogonal array

Orthogonal array with N runs, k factors, s symbols and strength t is a set of N k-tuples (called runs) with elements from $\{0,\dots ,s-1\}$ such that for every set of t coordinates every combination of symbols in this coordiantes appears equal numer of times across the runs. The common notion of such orthogonal array is $OA(N,k,s,t)$. It is easy to see, that N is divisible by $s^{t}$ — number of all possible symbol combinations in the t coordinates. The number ${\frac {N}{s^{t}}}$ is called an index of orthogonal array.

Statistical applications

Statistics is a primary application of orthogonal arrays. Experiments based on orthogonal arrays require less tests and yet provide a lot of info.

Particular cases

Some of mathematical constructions are particular cases of orthogonal arrays. For example, latin squares are $OA(n^{2},3,n,2)$. In order to see this, consider all triples $(i,j,s(i,j))$ where $s(i,j)$ — symbol in i-th row and j-th column in the latic square. Then $n^{2}$ such triples for all $i,j\in {0,\dots ,s-1}$ form an orthogonal array with strength 2: there is a single cell with given coordinates, single cell with given row and symbol in the cell and a single cell with given column and symbol in the cell. Here is a simple example:

Latin square	Orthogonal array
1 2 3 3 1 2 2 3 1	(1,1,1) (1,2,2) (1,3,3) (2,1,3) (2,2,1) (2,3,2) (3,1,2) (3,2,3) (3,3,1)

A set of k orthogonal latin square can be converted to $OA(n^{2},k,n,2)$ in a similar way.

Main results

The main result in theory of orthogonal arrays is the lower linear programming bound on the number of runs in the orthogonal array.